Abstract: According to the classical result of J. P. Serre ([S]) any two points on a closed Riemannian manifold can be connected by infinitely many geodesics.The length of a shortest of them trivially does not exceed the diameter d of the manifold.But how long are the shortest remaining geodesics?In this paper we prove that any two points on a closed n-dimensional Riemannian manifold can be connected by two distinct geodesics of length Ä 2qd Ä 2nd , where q is the smallest value of i such that the i th homotopy group of the manifold is nontrivial.